Variational quantum-neural hybrid imaginary time evolution

The gist: The proposed variational quantum-neural hybrid imaginary time evolution (VQNHITE) method enhances the expressive power of variational imaginary-time evolution (VITE) by incorporating a neural network to mitigate expressivity limitations, consistently achieving higher fidelity with exact ITE states compared to conventional VITE methods

Variational Imaginary Time Evolution (VITE)

Imaginary time evolution (ITE) is a powerful approach that leads to numerous applications, such as ground state search [1, 2] and the preparation of thermal equilibrium states [3] in a wide range of fields, including condensed matter physics, high-energy physics, and quantum chemistry [4–12]. The exact ITE dynamics are given by the operator e−Hβˆ, where Hˆ is the Hamiltonian. Since this operator is nonunitary, it cannot be directly realized on quantum devices. Variational ITE (VITE) approximates this dynamics using a parameterized quantum circuit updated through a hybrid quantum-classical loop. The state at imaginary time β is given by ψ(β)⟩ = exp(−Hβˆ)ψ(0)⟩q⟨ψ(0) exp(−2Hβˆ)ψ(0)⟩, which has unit norm. The evolution of the parameters is approximated using the Euler method with a small step size δβ, yielding θ(β + δβ) ≃ θ(β) + M−1(β) · C(β)δβ.

Variational Quantum-Neural Hybrid Eigensolver (VQNHE)

The variational quantum-neural hybrid eigensolver (VQNHE) enhances the expressivity by introducing a non-unitary operator ˆf(ϕ) generated by a classical neural network (NN). The trial quantum state is constructed as φ˜(θ, ϕ)⟩ = ˆf(ϕ)Uˆ(θ)¯0, where Uˆ(θ) is a unitary operator implemented by a parameterized quantum circuit, and ˆf(ϕ) is a non-unitary operator generated by the neural network. In VQNHE, expectation values of Hˆ are evaluated from a state vector modified by the non-unitary operator through post-processing.

Variational Quantum-Neural Hybrid ITE (VQNHITE)

The proposed VQNHITE approach incorporates the NN-based non-unitary transformation into variational ITE. The procedure consists of two stages: (i) preparing suitable initial parameters θ and ϕ at a small imaginary-time step β = δβ, and (ii) performing imaginary-time evolution for general β using a joint update of θ and ϕ. To mitigate the sensitivity of NN-based methods to initial parameters, the method introduces an initial-parameter optimization procedure at a small imaginary-time step, which stabilizes the subsequent variational evolution. The cost function Fcost = 1 − ⟨ψ(δβ)Φ(θ, ϕ)⟩2, where Φ(θ, ϕ)⟩ = C(θ, ϕ)φ˜(θ, ϕ)⟩ and C(θ, ϕ) = 1/p⟨φ˜(θ, ϕ)φ˜(θ, ϕ)⟩. The derivatives of Fcost with respect to θj and ϕj are given by complex expressions involving DϕjFcost and DθjFcost. All components of these derivatives can be computed with quantum devices as shown in Appendix A.

Parameter Updates for VQNHITE

The hybrid update equations generalize the VITE update equation so that the set of variational parameters consists of the combined variables ⟨θ, ϕ⟩. The generalized matrix elements Re ∂ϕjΦ(θ, ϕ)∂ϕk Φ(θ, ϕ) are given by a complex expression involving C and DϕjFcost and DϕkFcost. Similarly, the update for θj involves terms such as Re ∂θjUˆ† ˆf(ϕ)∂ϕk ˆf(ϕ)φ(θ)⟩. These matrix elements can be evaluated using the Hadamard test, as detailed in Appendix A.

Numerical Simulations and Results

Numerical simulations confirm that VQNHITE consistently achieves higher fidelity with the exact imaginary-time–evolved states than conventional VITE across different circuit connectivities. The performance gap increases with β, holding both under realistic connectivity constraints (nearest-neighbor interactions) and in an all-to-all interactions setting. The results show that the NN improves expressivity even when the circuit topology is limited.

Conclusion

The VQNHITE method augments the VITE with a neural-network–based nonunitary operation, enhancing the expressive power of the variational ansatz. By incorporating the NN, this framework enables a more accurate approximation of ITE under limited quantum resources. The method stabilizes subsequent imaginary-time updates by using a gradient-based procedure to initialize the NN parameters at a small imaginary-time step. Future work includes extending VQNHITE to larger scale systems and exploring alternative neural-network architectures. The hybrid strategy developed here provides a promising direction for accurate imaginary-time simulation on NISQ hardware.

How it works

1. Preparation of Parameters: At β = 0, parameters are initialized such that both the neural-network operator ˆfϕ(β=0) and the variational quantum circuit Uˆ(θ(β = 0)) reduce to the identity operator.

2. Initialization Optimization: The parameters θ and ϕ are optimized for β = δβ using the cost function Fcost = 1 − ⟨ψ(δβ)Φ(θ, ϕ)⟩2.

3. Hybrid Update: For general β > δβ, the parameters are updated using the joint update of θ and ϕ based on the generalized matrix elements derived in Eq. (28) and (30).

4. Evaluation of Derivatives: All components of the derivatives, including DϕjFcost and DθjFcost, can be computed with quantum devices as shown in Appendix A.

5. Final Fidelity Comparison: The method is verified by comparing the fidelity between variationally evolved states and the exact ITE state, showing consistently higher fidelity for VQNHITE.

Key Components of VQNHITE

**- Trial State Construction: The trial quantum state is defined as φ˜(θ, ϕ)⟩ = ˆf(ϕ)Uˆ(θ)¯0. 1- Non-unitary Operator: The operator ˆf(ϕ) is generated by a neural network, where fϕ(s) is computed by an NN. 1- Initial State Preparation: The initial state ψ(β=0)⟩ is chosen as the all-plus state + · · · +⟩. 1- Stabilization Step: Optimization of θ and ϕ for β = δβ stabilizes the subsequent variational evolution. The method utilizes a gradient-based procedure to initialize the NN parameters, which stabilizes the subsequent imaginary-time updates. The overall overhead remains manageable for near-term devices. The paper is organized as follows. In Sec. II, we present an overview of VITE. In Sec. III, we briefly review VQNHE. In Sec.

1. Variational quantum-neural hybrid imaginary time evolution (VQNHITE) can be used to track non-unitary dynamics with higher accuracy compared to conventional variational quantum algorithms because it enhances the expressive power of variational states and improves the estimation of expectation values.

2. The system can perform more accurate tracking of imaginary-time dynamics by combining a neural network with a parameterized quantum circuit, which is stated as enabling more accurate approximation of imaginary-time dynamics than standard VITE.

3. The improved system can mitigate the instability arising from randomly initialized neural network parameters by introducing an initial-parameter optimization procedure at a small imaginary-time step, which stabilizes the subsequent variational evolution.

4. The VQNHITE method allows for the virtual generation of states by optimizing parameters, as it does not physically obtain the normalized state (Phi⟩) but can virtually generate the state.

5. The system can be applied to solve problems in condensed matter physics, high-energy physics, and quantum chemistry by using imaginary time evolution for tasks such as ground state search [1, 2] and the preparation of thermal equilibrium states [3].

6. The VQNHITE framework can be extended to compute expectation values using the state it virtually produces by adopting the same approach as VQNHE, which allows for calculating energy estimates via post-processing.